LMFDB - Embedded newform 672.2.c.b.337.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(337,672)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("672.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$1.40961 - 0.114062i$ of defining polynomial
Character	$\chi$	$=$	672.337
Dual form			672.2.c.b.337.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.12875i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.12875i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.76717i q^{11} +0.456247i q^{13} +1.12875 q^{15} +0.415006 q^{17} +7.63843i q^{19} -1.00000i q^{21} -1.58499 q^{23} +3.72593 q^{25} -1.00000i q^{27} +6.72593i q^{29} +5.89592 q^{31} -4.76717 q^{33} +1.12875i q^{35} +5.89592i q^{37} -0.456247 q^{39} -0.415006 q^{41} -9.43967i q^{43} +1.12875i q^{45} -11.2769 q^{47} +1.00000 q^{49} +0.415006i q^{51} +7.63843i q^{53} +5.38093 q^{55} -7.63843 q^{57} -4.00000i q^{59} -1.80125i q^{61} +1.00000 q^{63} +0.514988 q^{65} +8.09467i q^{67} -1.58499i q^{69} -10.2068 q^{71} -3.34500 q^{73} +3.72593i q^{75} -4.76717i q^{77} +4.83001 q^{79} +1.00000 q^{81} -5.53434i q^{83} -0.468436i q^{85} -6.72593 q^{87} +4.92999 q^{89} -0.456247i q^{91} +5.89592i q^{93} +8.62185 q^{95} +16.4468 q^{97} -4.76717i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 - 8 * q^9	$ 8 q - 8 q^{7} - 8 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} - 24 q^{25} - 8 q^{31} + 12 q^{33} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 8 q^{63} - 16 q^{65} + 28 q^{71} - 8 q^{73} + 40 q^{79} + 8 q^{81} + 20 q^{89} - 40 q^{95} + 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 4 * q^15 + 4 * q^17 - 12 * q^23 - 24 * q^25 - 8 * q^31 + 12 * q^33 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^63 - 16 * q^65 + 28 * q^71 - 8 * q^73 + 40 * q^79 + 8 * q^81 + 20 * q^89 - 40 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 1.12875i	− 0.504791i	−0.967624	−	0.252395i	$-0.918782\pi$
$5$	− 1.12875i	− 0.504791i	0.967624	−	0.252395i	$-0.0812184\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.76717i	1.43736i	0.695343	+	0.718678i	$0.255253\pi$
$11$	4.76717i	1.43736i	−0.695343	+	0.718678i	$0.744747\pi$
$12$	0	0
$13$	0.456247i	0.126540i	0.997996	+	0.0632701i	$0.0201530\pi$
$13$	0.456247i	0.126540i	−0.997996	+	0.0632701i	$0.979847\pi$
$14$	0	0
$15$	1.12875	0.291441
$16$	0	0
$17$	0.415006	0.100654	0.0503268	−	0.998733i	$-0.483974\pi$
$17$	0.415006	0.100654	0.0503268	+	0.998733i	$0.483974\pi$
$18$	0	0
$19$	7.63843i	1.75237i	0.481970	+	0.876187i	$0.339921\pi$
$19$	7.63843i	1.75237i	−0.481970	+	0.876187i	$0.660079\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	−1.58499	−0.330494	−0.165247	−	0.986252i	$-0.552842\pi$
$23$	−1.58499	−0.330494	−0.165247	+	0.986252i	$0.552842\pi$
$24$	0	0
$25$	3.72593	0.745186
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.72593i	1.24897i	0.781035	+	0.624487i	$0.214692\pi$
$29$	6.72593i	1.24897i	−0.781035	+	0.624487i	$0.785308\pi$
$30$	0	0
$31$	5.89592	1.05894	0.529469	−	0.848329i	$-0.322391\pi$
$31$	5.89592	1.05894	0.529469	+	0.848329i	$0.322391\pi$
$32$	0	0
$33$	−4.76717	−0.829858
$34$	0	0
$35$	1.12875i	0.190793i
$36$	0	0
$37$	5.89592i	0.969283i	0.874713	+	0.484642i	$0.161050\pi$
$37$	5.89592i	0.969283i	−0.874713	+	0.484642i	$0.838950\pi$
$38$	0	0
$39$	−0.456247	−0.0730581
$40$	0	0
$41$	−0.415006	−0.0648130	−0.0324065	−	0.999475i	$-0.510317\pi$
$41$	−0.415006	−0.0648130	−0.0324065	+	0.999475i	$0.510317\pi$
$42$	0	0
$43$	− 9.43967i	− 1.43954i	−0.694214	−	0.719768i	$-0.744248\pi$
$43$	− 9.43967i	− 1.43954i	0.694214	−	0.719768i	$-0.255752\pi$
$44$	0	0
$45$	1.12875i	0.168264i
$46$	0	0
$47$	−11.2769	−1.64490	−0.822449	−	0.568839i	$-0.807393\pi$
$47$	−11.2769	−1.64490	−0.822449	+	0.568839i	$0.807393\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.415006i	0.0581124i
$52$	0	0
$53$	7.63843i	1.04922i	0.851343	+	0.524609i	$0.175789\pi$
$53$	7.63843i	1.04922i	−0.851343	+	0.524609i	$0.824211\pi$
$54$	0	0
$55$	5.38093	0.725565
$56$	0	0
$57$	−7.63843	−1.01173
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	− 1.80125i	− 0.230626i	−0.993329	−	0.115313i	$-0.963213\pi$
$61$	− 1.80125i	− 0.230626i	0.993329	−	0.115313i	$-0.0367871\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0.514988	0.0638764
$66$	0	0
$67$	8.09467i	0.988922i	0.869200	+	0.494461i	$0.164634\pi$
$67$	8.09467i	0.988922i	−0.869200	+	0.494461i	$0.835366\pi$
$68$	0	0
$69$	− 1.58499i	− 0.190811i
$70$	0	0
$71$	−10.2068	−1.21133	−0.605665	−	0.795720i	$-0.707093\pi$
$71$	−10.2068	−1.21133	−0.605665	+	0.795720i	$0.707093\pi$
$72$	0	0
$73$	−3.34500	−0.391503	−0.195751	−	0.980654i	$-0.562715\pi$
$73$	−3.34500	−0.391503	−0.195751	+	0.980654i	$0.562715\pi$
$74$	0	0
$75$	3.72593i	0.430233i
$76$	0	0
$77$	− 4.76717i	− 0.543270i
$78$	0	0
$79$	4.83001	0.543419	0.271709	−	0.962379i	$-0.412411\pi$
$79$	4.83001	0.543419	0.271709	+	0.962379i	$0.412411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.53434i	− 0.607473i	−0.952756	−	0.303737i	$-0.901766\pi$
$83$	− 5.53434i	− 0.607473i	0.952756	−	0.303737i	$-0.0982343\pi$
$84$	0	0
$85$	− 0.468436i	− 0.0508090i
$86$	0	0
$87$	−6.72593	−0.721095
$88$	0	0
$89$	4.92999	0.522578	0.261289	−	0.965261i	$-0.415852\pi$
$89$	4.92999	0.522578	0.261289	+	0.965261i	$0.415852\pi$
$90$	0	0
$91$	− 0.456247i	− 0.0478277i
$92$	0	0
$93$	5.89592i	0.611378i
$94$	0	0
$95$	8.62185	0.884583
$96$	0	0
$97$	16.4468	1.66992	0.834962	−	0.550308i	$-0.185490\pi$
$97$	16.4468	1.66992	0.834962	+	0.550308i	$0.185490\pi$
$98$	0	0
$99$	− 4.76717i	− 0.479119i
$100$	0	0
$101$	− 16.4056i	− 1.63242i	−0.577757	−	0.816209i	$-0.696072\pi$
$101$	− 16.4056i	− 1.63242i	0.577757	−	0.816209i	$-0.303928\pi$
$102$	0	0
$103$	17.1728	1.69208	0.846042	−	0.533117i	$-0.178979\pi$
$103$	17.1728	1.69208	0.846042	+	0.533117i	$0.178979\pi$
$104$	0	0
$105$	−1.12875	−0.110154
$106$	0	0
$107$	− 12.7672i	− 1.23425i	−0.786865	−	0.617125i	$-0.788297\pi$
$107$	− 12.7672i	− 1.23425i	0.786865	−	0.617125i	$-0.211703\pi$
$108$	0	0
$109$	7.27685i	0.696996i	0.937310	+	0.348498i	$0.113308\pi$
$109$	7.27685i	0.696996i	−0.937310	+	0.348498i	$0.886692\pi$
$110$	0	0
$111$	−5.89592	−0.559616
$112$	0	0
$113$	−3.34500	−0.314671	−0.157336	−	0.987545i	$-0.550290\pi$
$113$	−3.34500	−0.314671	−0.157336	+	0.987545i	$0.550290\pi$
$114$	0	0
$115$	1.78906i	0.166830i
$116$	0	0
$117$	− 0.456247i	− 0.0421801i
$118$	0	0
$119$	−0.415006	−0.0380435
$120$	0	0
$121$	−11.7259	−1.06599
$122$	0	0
$123$	− 0.415006i	− 0.0374198i
$124$	0	0
$125$	− 9.84937i	− 0.880954i
$126$	0	0
$127$	10.4468	0.927007	0.463504	−	0.886095i	$-0.346592\pi$
$127$	10.4468	0.927007	0.463504	+	0.886095i	$0.346592\pi$
$128$	0	0
$129$	9.43967	0.831117
$130$	0	0
$131$	6.25749i	0.546720i	0.961912	+	0.273360i	$0.0881350\pi$
$131$	6.25749i	0.546720i	−0.961912	+	0.273360i	$0.911865\pi$
$132$	0	0
$133$	− 7.63843i	− 0.662335i
$134$	0	0
$135$	−1.12875	−0.0971471
$136$	0	0
$137$	−12.9618	−1.10740	−0.553702	−	0.832715i	$-0.686785\pi$
$137$	−12.9618	−1.10740	−0.553702	+	0.832715i	$0.686785\pi$
$138$	0	0
$139$	− 18.3644i	− 1.55764i	−0.627245	−	0.778822i	$-0.715817\pi$
$139$	− 18.3644i	− 1.55764i	0.627245	−	0.778822i	$-0.284183\pi$
$140$	0	0
$141$	− 11.2769i	− 0.949682i
$142$	0	0
$143$	−2.17501	−0.181883
$144$	0	0
$145$	7.59187	0.630471
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	15.6384i	1.28115i	0.767896	+	0.640575i	$0.221304\pi$
$149$	15.6384i	1.28115i	−0.767896	+	0.640575i	$0.778696\pi$
$150$	0	0
$151$	−15.2769	−1.24321	−0.621606	−	0.783330i	$-0.713520\pi$
$151$	−15.2769	−1.24321	−0.621606	+	0.783330i	$0.713520\pi$
$152$	0	0
$153$	−0.415006	−0.0335512
$154$	0	0
$155$	− 6.65500i	− 0.534543i
$156$	0	0
$157$	− 21.7824i	− 1.73843i	−0.494437	−	0.869214i	$-0.664626\pi$
$157$	− 21.7824i	− 1.73843i	0.494437	−	0.869214i	$-0.335374\pi$
$158$	0	0
$159$	−7.63843	−0.605767
$160$	0	0
$161$	1.58499	0.124915
$162$	0	0
$163$	− 4.60966i	− 0.361056i	−0.983570	−	0.180528i	$-0.942219\pi$
$163$	− 4.60966i	− 0.361056i	0.983570	−	0.180528i	$-0.0577807\pi$
$164$	0	0
$165$	5.38093i	0.418905i
$166$	0	0
$167$	22.9618	1.77684	0.888420	−	0.459032i	$-0.151804\pi$
$167$	22.9618	1.77684	0.888420	+	0.459032i	$0.151804\pi$
$168$	0	0
$169$	12.7918	0.983988
$170$	0	0
$171$	− 7.63843i	− 0.584125i
$172$	0	0
$173$	0.216252i	0.0164413i	0.999966	+	0.00822067i	$0.00261675\pi$
$173$	0.216252i	0.0164413i	−0.999966	+	0.00822067i	$0.997383\pi$
$174$	0	0
$175$	−3.72593	−0.281654
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	22.6165i	1.69044i	0.534419	+	0.845220i	$0.320530\pi$
$179$	22.6165i	1.69044i	−0.534419	+	0.845220i	$0.679470\pi$
$180$	0	0
$181$	− 7.73310i	− 0.574797i	−0.957811	−	0.287398i	$-0.907210\pi$
$181$	− 7.73310i	− 0.574797i	0.957811	−	0.287398i	$-0.0927903\pi$
$182$	0	0
$183$	1.80125	0.133152
$184$	0	0
$185$	6.65500	0.489285
$186$	0	0
$187$	1.97840i	0.144675i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−10.2068	−0.738541	−0.369271	−	0.929322i	$-0.620392\pi$
$191$	−10.2068	−0.738541	−0.369271	+	0.929322i	$0.620392\pi$
$192$	0	0
$193$	5.96407	0.429303	0.214651	−	0.976691i	$-0.431138\pi$
$193$	5.96407	0.429303	0.214651	+	0.976691i	$0.431138\pi$
$194$	0	0
$195$	0.514988i	0.0368791i
$196$	0	0
$197$	− 18.8084i	− 1.34004i	−0.742341	−	0.670022i	$-0.766285\pi$
$197$	− 18.8084i	− 1.34004i	0.742341	−	0.670022i	$-0.233715\pi$
$198$	0	0
$199$	−3.27685	−0.232290	−0.116145	−	0.993232i	$-0.537054\pi$
$199$	−3.27685	−0.232290	−0.116145	+	0.993232i	$0.537054\pi$
$200$	0	0
$201$	−8.09467	−0.570954
$202$	0	0
$203$	− 6.72593i	− 0.472068i
$204$	0	0
$205$	0.468436i	0.0327170i
$206$	0	0
$207$	1.58499	0.110165
$208$	0	0
$209$	−36.4137	−2.51879
$210$	0	0
$211$	10.1628i	0.699637i	0.936817	+	0.349819i	$0.113757\pi$
$211$	10.1628i	0.699637i	−0.936817	+	0.349819i	$0.886243\pi$
$212$	0	0
$213$	− 10.2068i	− 0.699361i
$214$	0	0
$215$	−10.6550	−0.726665
$216$	0	0
$217$	−5.89592	−0.400241
$218$	0	0
$219$	− 3.34500i	− 0.226034i
$220$	0	0
$221$	0.189345i	0.0127367i
$222$	0	0
$223$	−21.9668	−1.47101	−0.735504	−	0.677520i	$-0.763055\pi$
$223$	−21.9668	−1.47101	−0.735504	+	0.677520i	$0.763055\pi$
$224$	0	0
$225$	−3.72593	−0.248395
$226$	0	0
$227$	7.91752i	0.525504i	0.964863	+	0.262752i	$0.0846301\pi$
$227$	7.91752i	0.525504i	−0.964863	+	0.262752i	$0.915370\pi$
$228$	0	0
$229$	13.1606i	0.869676i	0.900509	+	0.434838i	$0.143194\pi$
$229$	13.1606i	0.869676i	−0.900509	+	0.434838i	$0.856806\pi$
$230$	0	0
$231$	4.76717	0.313657
$232$	0	0
$233$	17.2769	1.13184	0.565922	−	0.824459i	$-0.308520\pi$
$233$	17.2769	1.13184	0.565922	+	0.824459i	$0.308520\pi$
$234$	0	0
$235$	12.7287i	0.830330i
$236$	0	0
$237$	4.83001i	0.313743i
$238$	0	0
$239$	10.5219	0.680603	0.340301	−	0.940316i	$-0.389471\pi$
$239$	10.5219	0.680603	0.340301	+	0.940316i	$0.389471\pi$
$240$	0	0
$241$	−6.10686	−0.393378	−0.196689	−	0.980466i	$-0.563019\pi$
$241$	−6.10686	−0.393378	−0.196689	+	0.980466i	$0.563019\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 1.12875i	− 0.0721130i
$246$	0	0
$247$	−3.48501	−0.221746
$248$	0	0
$249$	5.53434	0.350725
$250$	0	0
$251$	− 19.8743i	− 1.25446i	−0.778836	−	0.627228i	$-0.784189\pi$
$251$	− 19.8743i	− 1.25446i	0.778836	−	0.627228i	$-0.215811\pi$
$252$	0	0
$253$	− 7.55594i	− 0.475038i
$254$	0	0
$255$	0.468436	0.0293346
$256$	0	0
$257$	−14.7550	−0.920391	−0.460195	−	0.887818i	$-0.652221\pi$
$257$	−14.7550	−0.920391	−0.460195	+	0.887818i	$0.652221\pi$
$258$	0	0
$259$	− 5.89592i	− 0.366355i
$260$	0	0
$261$	− 6.72593i	− 0.416325i
$262$	0	0
$263$	15.7600	0.971804	0.485902	−	0.874013i	$-0.338491\pi$
$263$	15.7600	0.971804	0.485902	+	0.874013i	$0.338491\pi$
$264$	0	0
$265$	8.62185	0.529636
$266$	0	0
$267$	4.92999i	0.301711i
$268$	0	0
$269$	21.8331i	1.33119i	0.746315	+	0.665593i	$0.231821\pi$
$269$	21.8331i	1.33119i	−0.746315	+	0.665593i	$0.768179\pi$
$270$	0	0
$271$	18.8328	1.14401	0.572005	−	0.820250i	$-0.306166\pi$
$271$	18.8328	1.14401	0.572005	+	0.820250i	$0.306166\pi$
$272$	0	0
$273$	0.456247	0.0276134
$274$	0	0
$275$	17.7622i	1.07110i
$276$	0	0
$277$	24.4684i	1.47017i	0.677977	+	0.735083i	$0.262857\pi$
$277$	24.4684i	1.47017i	−0.677977	+	0.735083i	$0.737143\pi$
$278$	0	0
$279$	−5.89592	−0.352979
$280$	0	0
$281$	14.5150	0.865892	0.432946	−	0.901420i	$-0.357474\pi$
$281$	14.5150	0.865892	0.432946	+	0.901420i	$0.357474\pi$
$282$	0	0
$283$	− 16.5509i	− 0.983850i	−0.870638	−	0.491925i	$-0.836293\pi$
$283$	− 16.5509i	− 0.983850i	0.870638	−	0.491925i	$-0.163707\pi$
$284$	0	0
$285$	8.62185i	0.510714i
$286$	0	0
$287$	0.415006	0.0244970
$288$	0	0
$289$	−16.8278	−0.989869
$290$	0	0
$291$	16.4468i	0.964131i
$292$	0	0
$293$	− 9.44377i	− 0.551711i	−0.961199	−	0.275855i	$-0.911039\pi$
$293$	− 9.44377i	− 0.551711i	0.961199	−	0.275855i	$-0.0889611\pi$
$294$	0	0
$295$	−4.51499	−0.262873
$296$	0	0
$297$	4.76717	0.276619
$298$	0	0
$299$	− 0.723150i	− 0.0418208i
$300$	0	0
$301$	9.43967i	0.544094i
$302$	0	0
$303$	16.4056	0.942477
$304$	0	0
$305$	−2.03315	−0.116418
$306$	0	0
$307$	0.361575i	0.0206362i	0.999947	+	0.0103181i	$0.00328441\pi$
$307$	0.361575i	0.0206362i	−0.999947	+	0.0103181i	$0.996716\pi$
$308$	0	0
$309$	17.1728i	0.976925i
$310$	0	0
$311$	11.2769	0.639452	0.319726	−	0.947510i	$-0.396409\pi$
$311$	11.2769	0.639452	0.319726	+	0.947510i	$0.396409\pi$
$312$	0	0
$313$	25.5837	1.44607	0.723037	−	0.690809i	$-0.242745\pi$
$313$	25.5837	1.44607	0.723037	+	0.690809i	$0.242745\pi$
$314$	0	0
$315$	− 1.12875i	− 0.0635977i
$316$	0	0
$317$	− 0.361575i	− 0.0203081i	−0.999948	−	0.0101540i	$-0.996768\pi$
$317$	− 0.361575i	− 0.0203081i	0.999948	−	0.0101540i	$-0.00323218\pi$
$318$	0	0
$319$	−32.0637	−1.79522
$320$	0	0
$321$	12.7672	0.712594
$322$	0	0
$323$	3.16999i	0.176383i
$324$	0	0
$325$	1.69995i	0.0942960i
$326$	0	0
$327$	−7.27685	−0.402411
$328$	0	0
$329$	11.2769	0.621713
$330$	0	0
$331$	7.80403i	0.428948i	0.976730	+	0.214474i	$0.0688037\pi$
$331$	7.80403i	0.428948i	−0.976730	+	0.214474i	$0.931196\pi$
$332$	0	0
$333$	− 5.89592i	− 0.323094i
$334$	0	0
$335$	9.13684	0.499199
$336$	0	0
$337$	8.76186	0.477289	0.238645	−	0.971107i	$-0.423297\pi$
$337$	8.76186	0.477289	0.238645	+	0.971107i	$0.423297\pi$
$338$	0	0
$339$	− 3.34500i	− 0.181675i
$340$	0	0
$341$	28.1069i	1.52207i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.78906	−0.0963196
$346$	0	0
$347$	2.07717i	0.111509i	0.998445	+	0.0557543i	$0.0177563\pi$
$347$	2.07717i	0.111509i	−0.998445	+	0.0557543i	$0.982244\pi$
$348$	0	0
$349$	18.7381i	1.00303i	0.865149	+	0.501514i	$0.167224\pi$
$349$	18.7381i	1.00303i	−0.865149	+	0.501514i	$0.832776\pi$
$350$	0	0
$351$	0.456247	0.0243527
$352$	0	0
$353$	−13.2450	−0.704961	−0.352481	−	0.935819i	$-0.614662\pi$
$353$	−13.2450	−0.704961	−0.352481	+	0.935819i	$0.614662\pi$
$354$	0	0
$355$	11.5209i	0.611468i
$356$	0	0
$357$	− 0.415006i	− 0.0219644i
$358$	0	0
$359$	5.69186	0.300405	0.150202	−	0.988655i	$-0.452007\pi$
$359$	5.69186	0.300405	0.150202	+	0.988655i	$0.452007\pi$
$360$	0	0
$361$	−39.3455	−2.07082
$362$	0	0
$363$	− 11.7259i	− 0.615452i
$364$	0	0
$365$	3.77566i	0.197627i
$366$	0	0
$367$	−11.2769	−0.588647	−0.294323	−	0.955706i	$-0.595094\pi$
$367$	−11.2769	−0.588647	−0.294323	+	0.955706i	$0.595094\pi$
$368$	0	0
$369$	0.415006	0.0216043
$370$	0	0
$371$	− 7.63843i	− 0.396567i
$372$	0	0
$373$	7.08751i	0.366977i	0.983022	+	0.183489i	$0.0587390\pi$
$373$	7.08751i	0.366977i	−0.983022	+	0.183489i	$0.941261\pi$
$374$	0	0
$375$	9.84937	0.508619
$376$	0	0
$377$	−3.06869	−0.158046
$378$	0	0
$379$	− 6.67781i	− 0.343016i	−0.985183	−	0.171508i	$-0.945136\pi$
$379$	− 6.67781i	− 0.343016i	0.985183	−	0.171508i	$-0.0548639\pi$
$380$	0	0
$381$	10.4468i	0.535208i
$382$	0	0
$383$	18.6550	0.953226	0.476613	−	0.879113i	$-0.341864\pi$
$383$	18.6550	0.953226	0.476613	+	0.879113i	$0.341864\pi$
$384$	0	0
$385$	−5.38093	−0.274238
$386$	0	0
$387$	9.43967i	0.479845i
$388$	0	0
$389$	20.3428i	1.03142i	0.856764	+	0.515709i	$0.172472\pi$
$389$	20.3428i	1.03142i	−0.856764	+	0.515709i	$0.827528\pi$
$390$	0	0
$391$	−0.657782	−0.0332654
$392$	0	0
$393$	−6.25749	−0.315649
$394$	0	0
$395$	− 5.45186i	− 0.274313i
$396$	0	0
$397$	10.9031i	0.547210i	0.961842	+	0.273605i	$0.0882161\pi$
$397$	10.9031i	0.547210i	−0.961842	+	0.273605i	$0.911784\pi$
$398$	0	0
$399$	7.63843	0.382400
$400$	0	0
$401$	29.1756	1.45696	0.728479	−	0.685068i	$-0.240228\pi$
$401$	29.1756	1.45696	0.728479	+	0.685068i	$0.240228\pi$
$402$	0	0
$403$	2.69000i	0.133998i
$404$	0	0
$405$	− 1.12875i	− 0.0560879i
$406$	0	0
$407$	−28.1069	−1.39321
$408$	0	0
$409$	−31.7237	−1.56864	−0.784318	−	0.620359i	$-0.786987\pi$
$409$	−31.7237	−1.56864	−0.784318	+	0.620359i	$0.786987\pi$
$410$	0	0
$411$	− 12.9618i	− 0.639360i
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	−6.24687	−0.306647
$416$	0	0
$417$	18.3644	0.899306
$418$	0	0
$419$	19.2769i	0.941736i	0.882204	+	0.470868i	$0.156059\pi$
$419$	19.2769i	0.941736i	−0.882204	+	0.470868i	$0.843941\pi$
$420$	0	0
$421$	− 19.3234i	− 0.941765i	−0.882196	−	0.470882i	$-0.843936\pi$
$421$	− 19.3234i	− 0.941765i	0.882196	−	0.470882i	$-0.156064\pi$
$422$	0	0
$423$	11.2769	0.548299
$424$	0	0
$425$	1.54628	0.0750057
$426$	0	0
$427$	1.80125i	0.0871684i
$428$	0	0
$429$	− 2.17501i	− 0.105010i
$430$	0	0
$431$	−14.4150	−0.694346	−0.347173	−	0.937801i	$-0.612858\pi$
$431$	−14.4150	−0.694346	−0.347173	+	0.937801i	$0.612858\pi$
$432$	0	0
$433$	−9.27685	−0.445817	−0.222908	−	0.974839i	$-0.571555\pi$
$433$	−9.27685	−0.445817	−0.222908	+	0.974839i	$0.571555\pi$
$434$	0	0
$435$	7.59187i	0.364002i
$436$	0	0
$437$	− 12.1069i	− 0.579150i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	15.6465i	0.743388i	0.928355	+	0.371694i	$0.121223\pi$
$443$	15.6465i	0.743388i	−0.928355	+	0.371694i	$0.878777\pi$
$444$	0	0
$445$	− 5.56472i	− 0.263793i
$446$	0	0
$447$	−15.6384	−0.739672
$448$	0	0
$449$	−12.6550	−0.597226	−0.298613	−	0.954374i	$-0.596524\pi$
$449$	−12.6550	−0.597226	−0.298613	+	0.954374i	$0.596524\pi$
$450$	0	0
$451$	− 1.97840i	− 0.0931594i
$452$	0	0
$453$	− 15.2769i	− 0.717769i
$454$	0	0
$455$	−0.514988	−0.0241430
$456$	0	0
$457$	−23.9237	−1.11910	−0.559551	−	0.828796i	$-0.689026\pi$
$457$	−23.9237	−1.11910	−0.559551	+	0.828796i	$0.689026\pi$
$458$	0	0
$459$	− 0.415006i	− 0.0193708i
$460$	0	0
$461$	− 6.74558i	− 0.314173i	−0.987585	−	0.157086i	$-0.949790\pi$
$461$	− 6.74558i	− 0.314173i	0.987585	−	0.157086i	$-0.0502101\pi$
$462$	0	0
$463$	11.1700	0.519113	0.259557	−	0.965728i	$-0.416424\pi$
$463$	11.1700	0.519113	0.259557	+	0.965728i	$0.416424\pi$
$464$	0	0
$465$	6.65500	0.308618
$466$	0	0
$467$	3.70935i	0.171648i	0.996310	+	0.0858242i	$0.0273523\pi$
$467$	3.70935i	0.171648i	−0.996310	+	0.0858242i	$0.972648\pi$
$468$	0	0
$469$	− 8.09467i	− 0.373777i
$470$	0	0
$471$	21.7824	1.00368
$472$	0	0
$473$	45.0005	2.06913
$474$	0	0
$475$	28.4602i	1.30585i
$476$	0	0
$477$	− 7.63843i	− 0.349739i
$478$	0	0
$479$	2.17501	0.0993788	0.0496894	−	0.998765i	$-0.484177\pi$
$479$	2.17501	0.0993788	0.0496894	+	0.998765i	$0.484177\pi$
$480$	0	0
$481$	−2.69000	−0.122653
$482$	0	0
$483$	1.58499i	0.0721197i
$484$	0	0
$485$	− 18.5643i	− 0.842962i
$486$	0	0
$487$	4.51499	0.204594	0.102297	−	0.994754i	$-0.467381\pi$
$487$	4.51499	0.204594	0.102297	+	0.994754i	$0.467381\pi$
$488$	0	0
$489$	4.60966	0.208456
$490$	0	0
$491$	− 1.91221i	− 0.0862967i	−0.999069	−	0.0431483i	$-0.986261\pi$
$491$	− 1.91221i	− 0.0862967i	0.999069	−	0.0431483i	$-0.0137388\pi$
$492$	0	0
$493$	2.79130i	0.125714i
$494$	0	0
$495$	−5.38093	−0.241855
$496$	0	0
$497$	10.2068	0.457840
$498$	0	0
$499$	27.4065i	1.22688i	0.789740	+	0.613442i	$0.210216\pi$
$499$	27.4065i	1.22688i	−0.789740	+	0.613442i	$0.789784\pi$
$500$	0	0
$501$	22.9618i	1.02586i
$502$	0	0
$503$	−17.1368	−0.764094	−0.382047	−	0.924143i	$-0.624781\pi$
$503$	−17.1368	−0.764094	−0.382047	+	0.924143i	$0.624781\pi$
$504$	0	0
$505$	−18.5178	−0.824030
$506$	0	0
$507$	12.7918i	0.568105i
$508$	0	0
$509$	− 21.6437i	− 0.959342i	−0.877449	−	0.479671i	$-0.840756\pi$
$509$	− 21.6437i	− 0.959342i	0.877449	−	0.479671i	$-0.159244\pi$
$510$	0	0
$511$	3.34500	0.147974
$512$	0	0
$513$	7.63843	0.337245
$514$	0	0
$515$	− 19.3837i	− 0.854148i
$516$	0	0
$517$	− 53.7587i	− 2.36430i
$518$	0	0
$519$	−0.216252	−0.00949241
$520$	0	0
$521$	28.3137	1.24045	0.620223	−	0.784426i	$-0.287042\pi$
$521$	28.3137	1.24045	0.620223	+	0.784426i	$0.287042\pi$
$522$	0	0
$523$	− 15.0687i	− 0.658908i	−0.944172	−	0.329454i	$-0.893135\pi$
$523$	− 15.0687i	− 0.658908i	0.944172	−	0.329454i	$-0.106865\pi$
$524$	0	0
$525$	− 3.72593i	− 0.162613i
$526$	0	0
$527$	2.44684	0.106586
$528$	0	0
$529$	−20.4878	−0.890774
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	− 0.189345i	− 0.00820145i
$534$	0	0
$535$	−14.4109	−0.623038
$536$	0	0
$537$	−22.6165	−0.975976
$538$	0	0
$539$	4.76717i	0.205337i
$540$	0	0
$541$	− 46.2990i	− 1.99055i	−0.0971017	−	0.995274i	$-0.530957\pi$
$541$	− 46.2990i	− 1.99055i	0.0971017	−	0.995274i	$-0.469043\pi$
$542$	0	0
$543$	7.73310	0.331859
$544$	0	0
$545$	8.21372	0.351837
$546$	0	0
$547$	2.03714i	0.0871020i	0.999051	+	0.0435510i	$0.0138671\pi$
$547$	2.03714i	0.0871020i	−0.999051	+	0.0435510i	$0.986133\pi$
$548$	0	0
$549$	1.80125i	0.0768753i
$550$	0	0
$551$	−51.3755	−2.18867
$552$	0	0
$553$	−4.83001	−0.205393
$554$	0	0
$555$	6.65500i	0.282489i
$556$	0	0
$557$	− 11.1234i	− 0.471315i	−0.971836	−	0.235658i	$-0.924276\pi$
$557$	− 11.1234i	− 0.471315i	0.971836	−	0.235658i	$-0.0757244\pi$
$558$	0	0
$559$	4.30683	0.182159
$560$	0	0
$561$	−1.97840	−0.0835282
$562$	0	0
$563$	− 37.2437i	− 1.56963i	−0.619727	−	0.784817i	$-0.712757\pi$
$563$	− 37.2437i	− 1.56963i	0.619727	−	0.784817i	$-0.287243\pi$
$564$	0	0
$565$	3.77566i	0.158843i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	22.9369	0.961564	0.480782	−	0.876840i	$-0.340353\pi$
$569$	22.9369	0.961564	0.480782	+	0.876840i	$0.340353\pi$
$570$	0	0
$571$	− 35.1634i	− 1.47154i	−0.677231	−	0.735770i	$-0.736820\pi$
$571$	− 35.1634i	− 1.47154i	0.677231	−	0.735770i	$-0.263180\pi$
$572$	0	0
$573$	− 10.2068i	− 0.426397i
$574$	0	0
$575$	−5.90558	−0.246280
$576$	0	0
$577$	13.7918	0.574162	0.287081	−	0.957906i	$-0.407315\pi$
$577$	13.7918	0.574162	0.287081	+	0.957906i	$0.407315\pi$
$578$	0	0
$579$	5.96407i	0.247858i
$580$	0	0
$581$	5.53434i	0.229603i
$582$	0	0
$583$	−36.4137	−1.50810
$584$	0	0
$585$	−0.514988	−0.0212921
$586$	0	0
$587$	26.9862i	1.11384i	0.830566	+	0.556920i	$0.188017\pi$
$587$	26.9862i	1.11384i	−0.830566	+	0.556920i	$0.811983\pi$
$588$	0	0
$589$	45.0355i	1.85566i
$590$	0	0
$591$	18.8084	0.773675
$592$	0	0
$593$	−31.8337	−1.30725	−0.653627	−	0.756817i	$-0.726753\pi$
$593$	−31.8337	−1.30725	−0.653627	+	0.756817i	$0.726753\pi$
$594$	0	0
$595$	0.468436i	0.0192040i
$596$	0	0
$597$	− 3.27685i	− 0.134113i
$598$	0	0
$599$	34.9006	1.42600	0.712999	−	0.701165i	$-0.247336\pi$
$599$	34.9006	1.42600	0.712999	+	0.701165i	$0.247336\pi$
$600$	0	0
$601$	−0.175010	−0.00713881	−0.00356941	−	0.999994i	$-0.501136\pi$
$601$	−0.175010	−0.00713881	−0.00356941	+	0.999994i	$0.501136\pi$
$602$	0	0
$603$	− 8.09467i	− 0.329641i
$604$	0	0
$605$	13.2356i	0.538104i
$606$	0	0
$607$	32.1705	1.30576	0.652881	−	0.757461i	$-0.273560\pi$
$607$	32.1705	1.30576	0.652881	+	0.757461i	$0.273560\pi$
$608$	0	0
$609$	6.72593	0.272548
$610$	0	0
$611$	− 5.14503i	− 0.208146i
$612$	0	0
$613$	− 8.37869i	− 0.338412i	−0.985581	−	0.169206i	$-0.945880\pi$
$613$	− 8.37869i	− 0.338412i	0.985581	−	0.169206i	$-0.0541203\pi$
$614$	0	0
$615$	−0.468436	−0.0188892
$616$	0	0
$617$	28.9618	1.16596	0.582980	−	0.812487i	$-0.301887\pi$
$617$	28.9618	1.16596	0.582980	+	0.812487i	$0.301887\pi$
$618$	0	0
$619$	− 6.85497i	− 0.275524i	−0.990465	−	0.137762i	$-0.956009\pi$
$619$	− 6.85497i	− 0.275524i	0.990465	−	0.137762i	$-0.0439910\pi$
$620$	0	0
$621$	1.58499i	0.0636036i
$622$	0	0
$623$	−4.92999	−0.197516
$624$	0	0
$625$	7.51221	0.300488
$626$	0	0
$627$	− 36.4137i	− 1.45422i
$628$	0	0
$629$	2.44684i	0.0975619i
$630$	0	0
$631$	40.2055	1.60056	0.800278	−	0.599629i	$-0.204685\pi$
$631$	40.2055	1.60056	0.800278	+	0.599629i	$0.204685\pi$
$632$	0	0
$633$	−10.1628	−0.403936
$634$	0	0
$635$	− 11.7918i	− 0.467945i
$636$	0	0
$637$	0.456247i	0.0180772i
$638$	0	0
$639$	10.2068	0.403777
$640$	0	0
$641$	−24.0737	−0.950854	−0.475427	−	0.879755i	$-0.657706\pi$
$641$	−24.0737	−0.950854	−0.475427	+	0.879755i	$0.657706\pi$
$642$	0	0
$643$	− 23.7559i	− 0.936841i	−0.883506	−	0.468421i	$-0.844823\pi$
$643$	− 23.7559i	− 0.936841i	0.883506	−	0.468421i	$-0.155177\pi$
$644$	0	0
$645$	− 10.6550i	− 0.419540i
$646$	0	0
$647$	32.6218	1.28250	0.641249	−	0.767333i	$-0.278417\pi$
$647$	32.6218	1.28250	0.641249	+	0.767333i	$0.278417\pi$
$648$	0	0
$649$	19.0687	0.748512
$650$	0	0
$651$	− 5.89592i	− 0.231079i
$652$	0	0
$653$	− 14.4109i	− 0.563942i	−0.959423	−	0.281971i	$-0.909012\pi$
$653$	− 14.4109i	− 0.563942i	0.959423	−	0.281971i	$-0.0909883\pi$
$654$	0	0
$655$	7.06313	0.275979
$656$	0	0
$657$	3.34500	0.130501
$658$	0	0
$659$	− 23.3315i	− 0.908866i	−0.890781	−	0.454433i	$-0.849842\pi$
$659$	− 23.3315i	− 0.908866i	0.890781	−	0.454433i	$-0.150158\pi$
$660$	0	0
$661$	44.1468i	1.71711i	0.512721	+	0.858555i	$0.328638\pi$
$661$	44.1468i	1.71711i	−0.512721	+	0.858555i	$0.671362\pi$
$662$	0	0
$663$	−0.189345	−0.00735356
$664$	0	0
$665$	−8.62185	−0.334341
$666$	0	0
$667$	− 10.6606i	− 0.412779i
$668$	0	0
$669$	− 21.9668i	− 0.849287i
$670$	0	0
$671$	8.58685	0.331492
$672$	0	0
$673$	11.6960	0.450846	0.225423	−	0.974261i	$-0.427624\pi$
$673$	11.6960	0.450846	0.225423	+	0.974261i	$0.427624\pi$
$674$	0	0
$675$	− 3.72593i	− 0.143411i
$676$	0	0
$677$	0.648756i	0.0249337i	0.999922	+	0.0124669i	$0.00396843\pi$
$677$	0.648756i	0.0249337i	−0.999922	+	0.0124669i	$0.996032\pi$
$678$	0	0
$679$	−16.4468	−0.631172
$680$	0	0
$681$	−7.91752	−0.303400
$682$	0	0
$683$	− 22.4909i	− 0.860589i	−0.902689	−	0.430294i	$-0.858410\pi$
$683$	− 22.4909i	− 0.860589i	0.902689	−	0.430294i	$-0.141590\pi$
$684$	0	0
$685$	14.6306i	0.559007i
$686$	0	0
$687$	−13.1606	−0.502107
$688$	0	0
$689$	−3.48501	−0.132768
$690$	0	0
$691$	− 45.2681i	− 1.72208i	−0.508538	−	0.861039i	$-0.669814\pi$
$691$	− 45.2681i	− 1.72208i	0.508538	−	0.861039i	$-0.330186\pi$
$692$	0	0
$693$	4.76717i	0.181090i
$694$	0	0
$695$	−20.7287	−0.786285
$696$	0	0
$697$	−0.172230	−0.00652366
$698$	0	0
$699$	17.2769i	0.653470i
$700$	0	0
$701$	− 1.03091i	− 0.0389370i	−0.999810	−	0.0194685i	$-0.993803\pi$
$701$	− 1.03091i	− 0.0389370i	0.999810	−	0.0194685i	$-0.00619740\pi$
$702$	0	0
$703$	−45.0355	−1.69855
$704$	0	0
$705$	−12.7287	−0.479391
$706$	0	0
$707$	16.4056i	0.616996i
$708$	0	0
$709$	− 10.5481i	− 0.396144i	−0.980188	−	0.198072i	$-0.936532\pi$
$709$	− 10.5481i	− 0.396144i	0.980188	−	0.198072i	$-0.0634679\pi$
$710$	0	0
$711$	−4.83001	−0.181140
$712$	0	0
$713$	−9.34500	−0.349973
$714$	0	0
$715$	2.45504i	0.0918131i
$716$	0	0
$717$	10.5219i	0.392946i
$718$	0	0
$719$	7.00502	0.261243	0.130622	−	0.991432i	$-0.458303\pi$
$719$	7.00502	0.261243	0.130622	+	0.991432i	$0.458303\pi$
$720$	0	0
$721$	−17.1728	−0.639547
$722$	0	0
$723$	− 6.10686i	− 0.227117i
$724$	0	0
$725$	25.0603i	0.930718i
$726$	0	0
$727$	17.8028	0.660270	0.330135	−	0.943934i	$-0.392906\pi$
$727$	17.8028	0.660270	0.330135	+	0.943934i	$0.392906\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.91752i	− 0.144895i
$732$	0	0
$733$	− 34.3300i	− 1.26801i	−0.773330	−	0.634004i	$-0.781410\pi$
$733$	− 34.3300i	− 1.26801i	0.773330	−	0.634004i	$-0.218590\pi$
$734$	0	0
$735$	1.12875	0.0416345
$736$	0	0
$737$	−38.5887	−1.42143
$738$	0	0
$739$	− 0.0946726i	− 0.00348259i	−0.999998	−	0.00174129i	$-0.999446\pi$
$739$	− 0.0946726i	− 0.00348259i	0.999998	−	0.00174129i	$-0.000554271\pi$
$740$	0	0
$741$	− 3.48501i	− 0.128025i
$742$	0	0
$743$	13.2755	0.487032	0.243516	−	0.969897i	$-0.421699\pi$
$743$	13.2755	0.487032	0.243516	+	0.969897i	$0.421699\pi$
$744$	0	0
$745$	17.6518	0.646713
$746$	0	0
$747$	5.53434i	0.202491i
$748$	0	0
$749$	12.7672i	0.466502i
$750$	0	0
$751$	−15.4187	−0.562637	−0.281318	−	0.959615i	$-0.590772\pi$
$751$	−15.4187	−0.562637	−0.281318	+	0.959615i	$0.590772\pi$
$752$	0	0
$753$	19.8743	0.724261
$754$	0	0
$755$	17.2437i	0.627562i
$756$	0	0
$757$	− 9.96685i	− 0.362251i	−0.983460	−	0.181126i	$-0.942026\pi$
$757$	− 9.96685i	− 0.362251i	0.983460	−	0.181126i	$-0.0579741\pi$
$758$	0	0
$759$	7.55594	0.274263
$760$	0	0
$761$	23.4837	0.851283	0.425642	−	0.904892i	$-0.360048\pi$
$761$	23.4837	0.851283	0.425642	+	0.904892i	$0.360048\pi$
$762$	0	0
$763$	− 7.27685i	− 0.263440i
$764$	0	0
$765$	0.468436i	0.0169363i
$766$	0	0
$767$	1.82499	0.0658966
$768$	0	0
$769$	22.9369	0.827125	0.413562	−	0.910476i	$-0.364284\pi$
$769$	22.9369	0.827125	0.413562	+	0.910476i	$0.364284\pi$
$770$	0	0
$771$	− 14.7550i	− 0.531388i
$772$	0	0
$773$	− 35.0956i	− 1.26230i	−0.775660	−	0.631150i	$-0.782583\pi$
$773$	− 35.0956i	− 1.26230i	0.775660	−	0.631150i	$-0.217417\pi$
$774$	0	0
$775$	21.9678	0.789106
$776$	0	0
$777$	5.89592	0.211515
$778$	0	0
$779$	− 3.16999i	− 0.113577i
$780$	0	0
$781$	− 48.6578i	− 1.74111i
$782$	0	0
$783$	6.72593	0.240365
$784$	0	0
$785$	−24.5869	−0.877542
$786$	0	0
$787$	17.9480i	0.639778i	0.947455	+	0.319889i	$0.103646\pi$
$787$	17.9480i	0.639778i	−0.947455	+	0.319889i	$0.896354\pi$
$788$	0	0
$789$	15.7600i	0.561071i
$790$	0	0
$791$	3.34500	0.118934
$792$	0	0
$793$	0.821814	0.0291835
$794$	0	0
$795$	8.62185i	0.305785i
$796$	0	0
$797$	34.4199i	1.21922i	0.792703	+	0.609608i	$0.208673\pi$
$797$	34.4199i	1.21922i	−0.792703	+	0.609608i	$0.791327\pi$
$798$	0	0
$799$	−4.67996	−0.165565
$800$	0	0
$801$	−4.92999	−0.174193
$802$	0	0
$803$	− 15.9462i	− 0.562729i
$804$	0	0
$805$	− 1.78906i	− 0.0630560i
$806$	0	0
$807$	−21.8331	−0.768561
$808$	0	0
$809$	1.89314	0.0665592	0.0332796	−	0.999446i	$-0.489405\pi$
$809$	1.89314	0.0665592	0.0332796	+	0.999446i	$0.489405\pi$
$810$	0	0
$811$	10.3400i	0.363086i	0.983383	+	0.181543i	$0.0581091\pi$
$811$	10.3400i	0.363086i	−0.983383	+	0.181543i	$0.941891\pi$
$812$	0	0
$813$	18.8328i	0.660495i
$814$	0	0
$815$	−5.20314	−0.182258
$816$	0	0
$817$	72.1042	2.52261
$818$	0	0
$819$	0.456247i	0.0159426i
$820$	0	0
$821$	− 2.40029i	− 0.0837706i	−0.999122	−	0.0418853i	$-0.986664\pi$
$821$	− 2.40029i	− 0.0837706i	0.999122	−	0.0418853i	$-0.0133364\pi$
$822$	0	0
$823$	−25.8655	−0.901616	−0.450808	−	0.892621i	$-0.648864\pi$
$823$	−25.8655	−0.901616	−0.450808	+	0.892621i	$0.648864\pi$
$824$	0	0
$825$	−17.7622	−0.618399
$826$	0	0
$827$	17.0928i	0.594375i	0.954819	+	0.297188i	$0.0960487\pi$
$827$	17.0928i	0.594375i	−0.954819	+	0.297188i	$0.903951\pi$
$828$	0	0
$829$	41.8893i	1.45488i	0.686174	+	0.727438i	$0.259289\pi$
$829$	41.8893i	1.45488i	−0.686174	+	0.727438i	$0.740711\pi$
$830$	0	0
$831$	−24.4684	−0.848801
$832$	0	0
$833$	0.415006	0.0143791
$834$	0	0
$835$	− 25.9181i	− 0.896933i
$836$	0	0
$837$	− 5.89592i	− 0.203793i
$838$	0	0
$839$	24.5500	0.847560	0.423780	−	0.905765i	$-0.360703\pi$
$839$	24.5500	0.847560	0.423780	+	0.905765i	$0.360703\pi$
$840$	0	0
$841$	−16.2381	−0.559936
$842$	0	0
$843$	14.5150i	0.499923i
$844$	0	0
$845$	− 14.4387i	− 0.496708i
$846$	0	0
$847$	11.7259	0.402908
$848$	0	0
$849$	16.5509	0.568026
$850$	0	0
$851$	− 9.34500i	− 0.320342i
$852$	0	0
$853$	38.7193i	1.32572i	0.748742	+	0.662862i	$0.230658\pi$
$853$	38.7193i	1.32572i	−0.748742	+	0.662862i	$0.769342\pi$
$854$	0	0
$855$	−8.62185	−0.294861
$856$	0	0
$857$	−45.2074	−1.54425	−0.772127	−	0.635468i	$-0.780807\pi$
$857$	−45.2074	−1.54425	−0.772127	+	0.635468i	$0.780807\pi$
$858$	0	0
$859$	57.1890i	1.95126i	0.219418	+	0.975631i	$0.429584\pi$
$859$	57.1890i	1.95126i	−0.219418	+	0.975631i	$0.570416\pi$
$860$	0	0
$861$	0.415006i	0.0141434i
$862$	0	0
$863$	45.1106	1.53558	0.767791	−	0.640701i	$-0.221356\pi$
$863$	45.1106	1.53558	0.767791	+	0.640701i	$0.221356\pi$
$864$	0	0
$865$	0.244094	0.00829944
$866$	0	0
$867$	− 16.8278i	− 0.571501i
$868$	0	0
$869$	23.0255i	0.781086i
$870$	0	0
$871$	−3.69317	−0.125138
$872$	0	0
$873$	−16.4468	−0.556641
$874$	0	0
$875$	9.84937i	0.332969i
$876$	0	0
$877$	17.6600i	0.596337i	0.954513	+	0.298168i	$0.0963757\pi$
$877$	17.6600i	0.596337i	−0.954513	+	0.298168i	$0.903624\pi$
$878$	0	0
$879$	9.44377	0.318530
$880$	0	0
$881$	6.78998	0.228760	0.114380	−	0.993437i	$-0.463512\pi$
$881$	6.78998	0.228760	0.114380	+	0.993437i	$0.463512\pi$
$882$	0	0
$883$	28.6015i	0.962516i	0.876579	+	0.481258i	$0.159820\pi$
$883$	28.6015i	0.962516i	−0.876579	+	0.481258i	$0.840180\pi$
$884$	0	0
$885$	− 4.51499i	− 0.151770i
$886$	0	0
$887$	−23.5919	−0.792138	−0.396069	−	0.918221i	$-0.629626\pi$
$887$	−23.5919	−0.792138	−0.396069	+	0.918221i	$0.629626\pi$
$888$	0	0
$889$	−10.4468	−0.350376
$890$	0	0
$891$	4.76717i	0.159706i
$892$	0	0
$893$	− 86.1374i	− 2.88248i
$894$	0	0
$895$	25.5284	0.853319
$896$	0	0
$897$	0.723150	0.0241453
$898$	0	0
$899$	39.6555i	1.32259i
$900$	0	0
$901$	3.16999i	0.105608i
$902$	0	0
$903$	−9.43967	−0.314133
$904$	0	0
$905$	−8.72871	−0.290152
$906$	0	0
$907$	2.02096i	0.0671050i	0.999437	+	0.0335525i	$0.0106821\pi$
$907$	2.02096i	0.0671050i	−0.999437	+	0.0335525i	$0.989318\pi$
$908$	0	0
$909$	16.4056i	0.544139i
$910$	0	0
$911$	−52.5524	−1.74114	−0.870569	−	0.492046i	$-0.836249\pi$
$911$	−52.5524	−1.74114	−0.870569	+	0.492046i	$0.836249\pi$
$912$	0	0
$913$	26.3832	0.873156
$914$	0	0
$915$	− 2.03315i	− 0.0672139i
$916$	0	0
$917$	− 6.25749i	− 0.206641i
$918$	0	0
$919$	−21.5237	−0.710002	−0.355001	−	0.934866i	$-0.615520\pi$
$919$	−21.5237	−0.710002	−0.355001	+	0.934866i	$0.615520\pi$
$920$	0	0
$921$	−0.361575	−0.0119143
$922$	0	0
$923$	− 4.65685i	− 0.153282i
$924$	0	0
$925$	21.9678i	0.722296i
$926$	0	0
$927$	−17.1728	−0.564028
$928$	0	0
$929$	−15.7500	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$929$	−15.7500	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$930$	0	0
$931$	7.63843i	0.250339i
$932$	0	0
$933$	11.2769i	0.369188i
$934$	0	0
$935$	2.23312	0.0730307
$936$	0	0
$937$	17.0687	0.557610	0.278805	−	0.960348i	$-0.410062\pi$
$937$	17.0687	0.557610	0.278805	+	0.960348i	$0.410062\pi$
$938$	0	0
$939$	25.5837i	0.834892i
$940$	0	0
$941$	26.5456i	0.865362i	0.901547	+	0.432681i	$0.142432\pi$
$941$	26.5456i	0.865362i	−0.901547	+	0.432681i	$0.857568\pi$
$942$	0	0
$943$	0.657782	0.0214203
$944$	0	0
$945$	1.12875	0.0367181
$946$	0	0
$947$	− 20.3265i	− 0.660522i	−0.943890	−	0.330261i	$-0.892863\pi$
$947$	− 20.3265i	− 0.660522i	0.943890	−	0.330261i	$-0.107137\pi$
$948$	0	0
$949$	− 1.52615i	− 0.0495408i
$950$	0	0
$951$	0.361575	0.0117249
$952$	0	0
$953$	−38.6855	−1.25315	−0.626573	−	0.779362i	$-0.715543\pi$
$953$	−38.6855	−1.25315	−0.626573	+	0.779362i	$0.715543\pi$
$954$	0	0
$955$	11.5209i	0.372809i
$956$	0	0
$957$	− 32.0637i	− 1.03647i
$958$	0	0
$959$	12.9618	0.418559
$960$	0	0
$961$	3.76186	0.121350
$962$	0	0
$963$	12.7672i	0.411416i
$964$	0	0
$965$	− 6.73192i	− 0.216708i
$966$	0	0
$967$	26.6606	0.857346	0.428673	−	0.903460i	$-0.358981\pi$
$967$	26.6606	0.857346	0.428673	+	0.903460i	$0.358981\pi$
$968$	0	0
$969$	−3.16999	−0.101835
$970$	0	0
$971$	− 3.27685i	− 0.105159i	−0.998617	−	0.0525796i	$-0.983256\pi$
$971$	− 3.27685i	− 0.105159i	0.998617	−	0.0525796i	$-0.0167443\pi$
$972$	0	0
$973$	18.3644i	0.588734i
$974$	0	0
$975$	−1.69995	−0.0544418
$976$	0	0
$977$	12.0387	0.385153	0.192576	−	0.981282i	$-0.438316\pi$
$977$	12.0387	0.385153	0.192576	+	0.981282i	$0.438316\pi$
$978$	0	0
$979$	23.5021i	0.751131i
$980$	0	0
$981$	− 7.27685i	− 0.232332i
$982$	0	0
$983$	−4.42188	−0.141036	−0.0705181	−	0.997510i	$-0.522465\pi$
$983$	−4.42188	−0.141036	−0.0705181	+	0.997510i	$0.522465\pi$
$984$	0	0
$985$	−21.2299	−0.676442
$986$	0	0
$987$	11.2769i	0.358946i
$988$	0	0
$989$	14.9618i	0.475758i
$990$	0	0
$991$	9.10184	0.289129	0.144565	−	0.989495i	$-0.453822\pi$
$991$	9.10184	0.289129	0.144565	+	0.989495i	$0.453822\pi$
$992$	0	0
$993$	−7.80403	−0.247653
$994$	0	0
$995$	3.69873i	0.117258i
$996$	0	0
$997$	− 42.8556i	− 1.35725i	−0.734485	−	0.678625i	$-0.762576\pi$
$997$	− 42.8556i	− 1.35725i	0.734485	−	0.678625i	$-0.237424\pi$
$998$	0	0
$999$	5.89592	0.186539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.c.b.337.6		8
3.2	odd	2		2016.2.c.e.1009.6		8
4.3	odd	2		168.2.c.b.85.2	yes	8
7.6	odd	2		4704.2.c.c.2353.3		8
8.3	odd	2		168.2.c.b.85.1	✓	8
8.5	even	2	inner	672.2.c.b.337.3		8
12.11	even	2		504.2.c.f.253.7		8
16.3	odd	4		5376.2.a.bp.1.2		4
16.5	even	4		5376.2.a.bq.1.3		4
16.11	odd	4		5376.2.a.bm.1.3		4
16.13	even	4		5376.2.a.bl.1.2		4
24.5	odd	2		2016.2.c.e.1009.3		8
24.11	even	2		504.2.c.f.253.8		8
28.27	even	2		1176.2.c.c.589.2		8
56.13	odd	2		4704.2.c.c.2353.6		8
56.27	even	2		1176.2.c.c.589.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.1	✓	8	8.3	odd	2
168.2.c.b.85.2	yes	8	4.3	odd	2
504.2.c.f.253.7		8	12.11	even	2
504.2.c.f.253.8		8	24.11	even	2
672.2.c.b.337.3		8	8.5	even	2	inner
672.2.c.b.337.6		8	1.1	even	1	trivial
1176.2.c.c.589.1		8	56.27	even	2
1176.2.c.c.589.2		8	28.27	even	2
2016.2.c.e.1009.3		8	24.5	odd	2
2016.2.c.e.1009.6		8	3.2	odd	2
4704.2.c.c.2353.3		8	7.6	odd	2
4704.2.c.c.2353.6		8	56.13	odd	2
5376.2.a.bl.1.2		4	16.13	even	4
5376.2.a.bm.1.3		4	16.11	odd	4
5376.2.a.bp.1.2		4	16.3	odd	4
5376.2.a.bq.1.3		4	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.12875i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.12875i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.76717i q^{11} +0.456247i q^{13} +1.12875 q^{15} +0.415006 q^{17} +7.63843i q^{19} -1.00000i q^{21} -1.58499 q^{23} +3.72593 q^{25} -1.00000i q^{27} +6.72593i q^{29} +5.89592 q^{31} -4.76717 q^{33} +1.12875i q^{35} +5.89592i q^{37} -0.456247 q^{39} -0.415006 q^{41} -9.43967i q^{43} +1.12875i q^{45} -11.2769 q^{47} +1.00000 q^{49} +0.415006i q^{51} +7.63843i q^{53} +5.38093 q^{55} -7.63843 q^{57} -4.00000i q^{59} -1.80125i q^{61} +1.00000 q^{63} +0.514988 q^{65} +8.09467i q^{67} -1.58499i q^{69} -10.2068 q^{71} -3.34500 q^{73} +3.72593i q^{75} -4.76717i q^{77} +4.83001 q^{79} +1.00000 q^{81} -5.53434i q^{83} -0.468436i q^{85} -6.72593 q^{87} +4.92999 q^{89} -0.456247i q^{91} +5.89592i q^{93} +8.62185 q^{95} +16.4468 q^{97} -4.76717i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 - 8 * q^9	\( 8 q - 8 q^{7} - 8 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} - 24 q^{25} - 8 q^{31} + 12 q^{33} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 8 q^{63} - 16 q^{65} + 28 q^{71} - 8 q^{73} + 40 q^{79} + 8 q^{81} + 20 q^{89} - 40 q^{95} + 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 4 * q^15 + 4 * q^17 - 12 * q^23 - 24 * q^25 - 8 * q^31 + 12 * q^33 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^63 - 16 * q^65 + 28 * q^71 - 8 * q^73 + 40 * q^79 + 8 * q^81 + 20 * q^89 - 40 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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